Note that, by definition,  \( {f_{C}^\uparrow }(z,v)\)  and \( {f_{C}^\downarrow }(z,v)\)  are the upper and lower support functions of the set \({\partial }_C f(z)\) .  Further, as shown in [1]}, \(\partial _C f(z)\)   for an LLC system is nonempty, convex and compact for all \(z \in \mathcal {Z}\) , and consequently, at each \(z \in \mathcal {Z}\) ,  the Clarke directional derivatives are bounded in each direction \(v\) . However, this does not hold in general for ELLC systems.  Nonetheless,  by the following proposition ,
the Clarke directional derivatives  in some specific directions are bounded from above or from below.
